On stationary and cycle-stationary sequences (Q1416116)

Let \(Z=(Z_k)\) be a stationary sequence with a state space \((E, {\mathcal E})\). Let \(A\in {\mathcal E}\) be such a set that the events \(\{Z_n\in A\}\) occur infinitely many times for both negative and positive \(n\). The times \(n\) when the events occur are called points. The points split the process into a two-sided sequence of cycles. The conditioning on \(Z_0\in A\) turns \(Z\) into a cycle-stationary sequence \(Z^{\circ}\). The point-at-zero duality means that \(Z^{\circ}\) behaves like \(Z\) conditioned on \(Z_0\in A\). The randomized-origin duality says that the cycle-stationary sequence behaves as a stationary one with origin shifted to a randomly chosen point. The two dualities coincide in the ergodic case. The author presents the main equivalence theorem on which the dualities rely.